Section 6.6B: More on Arctrig Functions.
Cancellation Properties of Trig/Arctrig Functions:
As stated before, trig and arctrig functions are inverses. However, due to the restrictive domains and ranges of the
arctrig functions, there are certain rules dealing with cancellation properties.
trig(arctrig(x)): When you have an expression of the form trig(arctrig(x)), you have the following situations.
• sin(arcsin(x)) = x if −1 ≤ x ≤ 1. Otherwise it is undefined.
• cos(arccos(x)) = x if −1 ≤ x ≤ 1. Otherwise it is undefined.
• tan(arctan(x)) = x for all values of x.
arctrig(trig(θ)): When you have an expression of the form arctrig(trig(θ)), what you have to do is work with the
domain of the arctrig function, with help from reference triangles. If θ is positive, then you need to label your triangle
in the first quadrant. If θ is negative, you label your triangle in the appropriate quadrant. (This is QIV for sine and
tangent, and QII for cosine.) Your final answer should be the angle in the proper quadrant.
Examples: Compute the following.
1.
4.
sin(arcsin(0.75))
4
arcsin sin − π
9
2.
5.
tan(arctan(3.1415))
15
arctan cos
π
8
3.
6.
cos(arccos(1.23))
5
arccos tan
π
9
More Word Problems involving ArcTrig Functions:
√
1. The triangle shown in the figure is a right triangle with one vertex on the graph y = 4 x. Express the angle
θ as a function of x.
2. Shown in the figure is the screen for a simple video arcade game in which ducks move from A to B at the rate
of 6 cm/sec. Bullets fired from point O travel 23 cm/sec. If a player shoots as soon as a duck appears at A,
at which angle φ should the gun be aimed in order to score a direct hit?
3. A conical paper cup has a radius of 3 inches. Find the degree measure of the angle β (see the figure) so that
the cone will have a volume of 60 in3 .
4. What is the angle of elevation of the sun α◦ , if the length of the shadow cast by a person h = 6.2 feet tall is
l = 8? Your answer should be a DEGREE measure, exact or rounded to the nearest whole number.
Trig Equations Revisited.
All our trig equations from Test 5 involved special triangles. Now we consider equations that doesn’t have that
restriction. Just like before, we will be working with acute reference angles, so it will be ideal to use values of the
form arctig(x), where x is a positive value.
Examples: Solve for all solutions on the interval [0, 2π) for the following equations.
1.
4.
9 sin2 (x) + 4 = 0
2 sin(x) = − cos(x)
2.
5.
13 cos2 (x) = 5
4 tan2 (x) + 4 tan(x) = −1
3.
6.
25 cos2 (x) − 10 cos(x) = −3
10 sin2 (x) − 33 sin(x) = 7